let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M,V holds f1 + f2 = f2 + f1
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,V holds f1 + f2 = f2 + f1
let f1, f2 be PartFunc of M,V; :: thesis: f1 + f2 = f2 + f1
A1: dom (f1 + f2) = (dom f2) /\ (dom f1) by VFUNCT_1:def 1
.= dom (f2 + f1) by VFUNCT_1:def 1 ;

now :: thesis: for x being Element of M st x in dom (f1 + f2) holds
(f1 + f2) /. x = (f2 + f1) /. x

let x be Element of M; :: thesis: ( x in dom (f1 + f2) implies (f1 + f2) /. x = (f2 + f1) /. x )
assume A2: x in dom (f1 + f2) ; :: thesis: (f1 + f2) /. x = (f2 + f1) /. x
hence (f1 + f2) /. x = (f2 /. x) + (f1 /. x) by VFUNCT_1:def 1
.= (f2 + f1) /. x by A1, A2, VFUNCT_1:def 1 ;
:: thesis: verum

end;

hence f1 + f2 = f2 + f1 by A1, PARTFUN2:1; :: thesis: verum